A SIMPLE GENERATIVE MODEL OF TRILOBITE SEGMENTATION AND GROWTH

Generative growth models have been the basis for numerous studies of morphological diversity and evolution. Most work has focused on modeling accretionary growth systems, with much less attention to discrete growth systems. Generative growth models for molting organisms, such as arthropods, have remained particularly elusive. However, our understanding of post-embryonic growth in trilobite species is sufficiently mature that it is now possible to model growth in a way that incorporates the addition of new parts as well as differential growth rates for existing parts across the trilobite body plan. Furthermore, body size data for a large sample of specimens of the trilobite species Aulacopleura koninckii (Barrande, 1846) make it possible to generate robust estimates for model parameters. Although the generative model described here was based on a relatively simple segmentation schedule, a diverse array of observed body sizes and relative proportions of body regions can be attained by altering only a few parameters at a time. Notably, small changes in growth rates can have large effects on body size (e.g. an increase of 4% increases body size by 350%). Subsampling of the empirical dataset indicates that parameters describing the growth gradient in the trunk are more sensitive to sample size than input parameters. Increasing the number of stages represented improves parameter estimates more quickly than increasing number of specimens per stage.


Introduction
Generative growth models have long attracted biologists and mathematicians. Some of the earliest work focused on the geometric form of coiled invertebrates, such as gastropods (e.g., Thompson 1942). In particular, the work by David Raup (Raup 1961, Raup 1966 was foundational for subsequent studies of mollusk growth and morphology using model-based descriptors, and as a result, mollusks have become a model group for studying accretionary growth, and theoretical modeling of morphology more generally (Urdy et al. 2010;Urdy 2015). Although generative models for other types of organisms have also been proposed (see McGhee 1999, McGhee 2015 for examples), arthropods have remained elusive, as "discrete growth systems are more difficult to model than accretionary growth systems. Perhaps the ultimate nightmare of any modeler of morphogenesis is organisms that molt, whose entire skeleton is episodically shed and regrown, often with the new skeleton possessing parts or elements not previously present in the organism" (McGhee 1999, p. 142).
Trilobites are an extinct group of marine arthropods with one of the best fossil records of any arthropod group. In part, this is due to the fact that the trilobite dorsal exoskeleton was highly impregnated with low-magnesium calcite (Wilmot and Fallick 1989). Because trilobites started biomineralizing the exoskeleton at very early post-embryonic stages, the growth and morphological development of many species can be pieced together from samples that include fossilized exuviae as well as individual body fossils (Hopkins 2017). With enough well-preserved specimens, it is possible to chart segmentation patterns and estimate growth rates. Segmentation and growth patterns in trilobites are now sufficiently well understood (e.g., Hughes 2003a, Hughes 2003b, Minelli et al. 2003, Hughes 2005, Hughes et al. 2006, Hughes 2007, Fusco et al. 2014, Fusco et al. 2016) that generative growth models are within reach. Further, measurements of body size and body proportions for a large sample of specimens are now available for one of the best studied trilobite species in this regard, Aulacopleura koninckii (Barrande, 1846) (Hughes et al. 2017). In this paper, I use these studies as a basis for developing a simple generative model for trilobites and explore the impact of varying different parameters on body size and body proportions.

Segmentation patterns and growth during trilobite ontogeny
Like some centipedes and most crustaceans, the final segmental composition of the trilobite body was attained during post-embryonic ontogeny through a series of molts . Trilobites had a hemianamorphic mode of segmentation, meaning they underwent a phase of molts during which new segments were added to the body followed by a phase during which molting continued without further increase in the number of body segments (Minelli et al. 2003, Minelli and. In trilobites, the earliest biomineralized molt was a single articulated sclerite comprised of some number of fused segments; this has been termed the protaspid period. The subsequent meraspid period began when molting resulted in the first articulation between the cephalon and the transitory pygidium, each comprised of fused segments. During subsequent molts, either new segments were generated near the posterior end of the transitory pygidium, or a new articulation appeared at the anterior end of the transitory pygidium (i.e., a thoracic tergite 1 was released into the thorax), or both. At some point, segment generation ceased (ending the anamorphic phase and initiating the epimorphic phase) and articulation generation ceased (ending the meraspid period and initiating the holaspid period) (Figure 1). The relative timing of these transitions varied across species, and three different schedules have been documented in trilobites: 1) hypoprotomeric, where individuals entered the epimorphic phase within the meraspid period (the cessation of new segment generation preceded the cessation of new thoracic articulation); 2) synanthromeric, where individuals entered the epimorphic phase and holaspid period synchronously; and 3) protarthrous, where individuals entered the epimorphic phases within the holaspid period (the cessation of new thoracic articulation preceded the cessation of new segment generation in the pygidium) ( Figure 1; Hughes et al. 2006).
During the meraspid period, the number of tergites in the thorax is a size-independent measure of relative age of the individuals. Per-molt size increase was constant for some sclerites for some portions of the ontogeny in at least some trilobites (Chatterton et al. 1990, Fusco et al. 2012), but detailed data-rich studies of the trilobite Aulocopleura koninckii indicate that the rate of growth in the thorax was governed by a growth field described by a continuous gradient rather than constant per-tergite rates (Fusco et al. 2014(Fusco et al. , 2016.

Description of the growth model
Because total body size at any given point during ontogeny (as well as potential maximum body size) was determined both by the segmentation schedule and the pattern of growth rates, several parameters are needed in order to model trilobite growth. Parameters are summarized in Table 1 and Figure 2. An R 1 Traditionally, the articulated sclerites in the thorax of the trilobite exoskeleton have been called "thoracic segments". Throughout this manuscript, I refer to the thoracic segments as "tergites" to reflect their anatomical similarity to other arthropods and distinguish them from body segments, which may not be positionally synchronous (Ortega-Hernández and Brena 2012)

Formatted: Highlight
Formatted: Highlight function for running the growth model using these parameters is available at www.github.com/meljh/trilogrowth.

Cephalic growth
Growth rate of cephalon (g.c). Although the relative size of features on the cephalon (e.g., the length of the glabella relative to the length of the cephalon) changes during ontogeny (Fusco et al. 2016, Hopkins and Pearson 2016, Hopkins 2017, this model assumes that the growth rate of the cephalon as a whole was constant (Chatterton et al. 1990, Fusco et al. 2012. For each molt M, the size of the cephalon is estimated as ce * g.c^M where ce is the initial size of the cephalon at M0 (see below) and g.c is the growth rate.
Defining the growth gradient of the trunk Fusco et al. (2014) found that an exponential decay function best described the distribution of per-molt local growth rates across the trunk during the meraspid period, specifically that the growth at relative position x along the trunk = a + b*exp (-λ(1-x)). When x = 0, this equation simplifies to growth rate = a; this is the lower bound for the growth gradient (lowM in Table 1, Figure 2). When x = 1, the equation simplifies to growth = a + b; this sum is the upper bound for the growth gradient (highM in Table 1, Figure 2). The exponential decay constant (λ) describes the concavity of the exponential decay curve, and is represented in the model by the parameter lambdaM (Table 1, Figure 2). Because the growth gradient may shift during the transition from the meraspid to holaspid stages (Fusco et al. 2016), three additional parameters are included to describe the holaspid growth gradient (lowH, highH, and lambdaH in Table 1, Figure 2). This gradient includes the pygidium, but newly released tergites were sourced from the anterior region of the growing transitory pygidium. Therefore the model effectively takes the following steps to determine the lengths of the tergites and transitory pygidium in each subsequent molt during the meraspid period: 1. Determine the relative position along the trunk of the median points of each tergite and on the transitory pygidium at the preceding molt.
2. Sample the exponential decay function at these relative positions to generate a vector of growth rates along the trunk.
3. Apply these rates to the respective lengths of each tergite and the transitory pygidium.
At this point the new length of the "pygidium" is the sum of the length of the new transitory pygidium and the length of the newly released tergite. Therefore the following additional steps are required: 4. Assign a length to the newly released tergite (see below). 5. Determine the length of the new transitory pygidium by subtracting the length of the newly released tergite.
For the holaspid period, the model follows steps 1 -3 for each molt.

Parameters determining initial input and transition points
Number of molts in lifespan (Molt). The number of molts determines the number of time-steps in the model. The only restriction on this parameter is that it must be greater than the number of molts during the meraspid period (Ter). The specimen modeled at Molt = 0 is conveniently the M0 stage (see Figure 1).
Number of molts during meraspid period (Ter). For most trilobite species, a thoracic tergite was released at each molt during the meraspid period. Thus this parameter will be equal to the terminal number of thoracic tergites for most species. The model currently does not accommodate exceptions (such as molts accompanied by no tergite release or multiple tergite release), although this could be modified in the future. In the model, when Molt < Ter, the parameters describing the growth gradient change are those defined for the meraspid period; when Molt ≥ Ter, the parameters describing the growth gradient are those defined for the holaspid period (see above).
Length of cephalon and transitory pygidium at M0 (ce and p, respectively). These parameters describe the size of the trilobite at the onset of epimorphic growth, where M0 refers to the molt characterized by one articulation between the cephalon and the transitory pygidium ( Figure 1, 2). These parameters set the initial size of the trilobite and can be estimated directly from fossil specimens or estimated from growth series.
Length of newly released tergite (t.p). This parameter determines the size of the tergite released from the transitory pygidium at each molt during meraspis ( Figure 2). Because the new tergite was sourced from the anterior region of the pygidium, the length of the newly-released tergite may be defined as a proportion of the pygidial length after the growth gradient has been applied. Number of molts during which initial or "meraspid" growth gradient is active (R). It is possible that the growth gradient did not always change at the start of holaspis. This parameter defines the number of molts for which the initial growth gradient was still active. It is thus constrained to range from zero to Molt. If R < Ter, then the change in growth rates occurs before the last tergite is released. If R > Ter, then the change in growth rates occurs after the last tergite is released. If R = Ter, then the transition occurs at the same time as the last tergite is released. If R = Molt, then there is no transition to a new growth gradient.
If the transition to the holaspid growth gradient coincided with the onset of epimorphosis, then the different segmentation schedules (Figure 1) could be implemented by varying this parameter appropriately. However, this is a hypothesis that has yet to be tested. It is also possible that the growth gradient changed gradually throughout ontogeny in some trilobites. This is not accommodated in the current model but could be potentially accommodated in the future by defining vectors of molt-dependent growth rates. Cephalic growth rate (g.c). The growth rate for both species was estimated using the slope of the OLS regression of logged cephalic lengths against the number of tergites in the thorax for all specimens certain to be meraspids 2 . For Aulacopleura koninckii, lines were fit to juvenile stages 9-17 (N = 137) because the number of specimens representing earlier stages were too few (N = 11), following Meraspid growth gradient. The number of tergites provides a size-independent measurement of age during the meraspid stage. For meraspids, model parameters a, b, and λ were estimated by fitting an exponential curve to average per-molt tergite growth rates against relative tergite position in the trunk.

Parameter estimates from empirical data
The relative position in the trunk of the posterior margin of each tergite was calculated by scaling the cumulative average lengths of the trunk sclerites to range from 0 to 1 and then finding the midpoint of 7 each tergite along this range. Growth rates between stages were calculated by dividing the average tergite lengths in stage i + 1 by the averages in stage i. For Aulacopleura koninckii, curves were fit to juvenile stages 9-17 (N = 137) because the number of specimens representing earlier stages were too few (N = 11), following Fusco et al. (2014). Including the earliest stages slightly alters the parameter estimates and the model output (Supplementary Figure 3).
Holaspid growth gradient. Unlike meraspids, holaspids are characterized by having the same number of tergites, so this trait cannot be used as a size-independent measure of age. However, because cephalic growth rates are assumed to be constant across ontogeny, holaspids can be assigned to molt stages by extrapolating from an OLS regression of ln(cephalic length) against molt stage from meraspid specimens (Fusco et al. 2016). Model parameters a, b, and λ are then estimated in the same fashion as for the meraspid growth gradient. Aulacopleura koninckii has a variable number of terminal segments, ranging from 18 to 22 and yielding 5 holaspid morphotypes. Note that A. koninckii provides an unusual case study for trilobites in having had a variable terminal number of thoracic tergites. For most species there appears to be no variation, and in some clades the terminal number of tergites was highly conserved (e.g., Lichidae species had 11 tergites at the onset of epimorphosis, Thomas andHolloway 1988, Whittington 2002). Fusco et al. (2016) found that growth gradients were similar among holaspid morphotypes; for this study, parameters were estimated using specimens with 20 tergites because this is the median terminal number of thoracic tergites achieved by A. koninckii individuals. Growth gradient curves are shown in Figure 3B.
Number of molts in lifespan (Molt). The total number of molts was set to the maximum estimated holaspid stage (see "Holaspid growth gradient" above).
Number of molts during meraspid period (Ter). The total number of molts during the meraspid stage was set to 20 for Aulacopleura koninckii, which is the median terminal number of thoracic tergites achieved by A. koninckii individuals.
Length of cephalon and transitory pygidium at M0 (ce and p, respectively). In order to estimate initial size parameters, the length of the cephalon at M0 was set as the intercept of the same regression used to determine the cephalic growth rate (see above and Fig. 1A). Because new tergites were released from the anterior part of the transitory pygidium during the meraspid phase, the average length of the pygidium stayed generally constant (Fig. 1C). Thus the length of the transitory pygidium at M0 was set as the average across all meraspids.

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Deleted: N Length of newly released tergite (t.p). This parameter was estimated as the average ratio between the length of newly released thoracic tergite in each meraspid molt (for all individuals with 17 or fewer tergites) relative to the sum of that length and that of the pygidium in the same molt (Fig. 1D). It is possible that some of the variation in this ratio across specimens (and that of pygidial length through ontogeny) is due to variation in the number of segments expressed in the transitory pygidium in different individuals with the same number of tergites. However, there is no consistent relationship between the number of segments in the transitory pygidium and its length (Supplementary Figure 2).
Number of molts during which initial or "meraspid" growth gradient is active (R). This parameter is set to equal the number of molts during the meraspid period (Ter), consistent with a trilobite that is synanthromeric.

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Deleted: Because Aulacopleura koninckii is synanthromeric, t Table 1. Parameters (all single variables) that can be adjusted in the model and parameter estimates for Aulacopleura koninckii (based on data from Hughes et al. 2017). An R function for calculating these estimates from the dataset is available at www.github.com/meljh/trilo-growth.

Output of the model for Aulacopleura koninckii
The output of the model is the estimated lengths of each dorsal sclerite at each molt. The output after applying the parameters calculated from measurements of Aulacopleura koninckii is shown in Figure 4.
Modeled body sizes and body proportions are similar to empirical data for A. koninckii (Table 2), and if the model is run multiple times, allowing the number of terminal thoracic tergites to vary between 17 and 22, the resulting total body sizes fall are consistent with the variation in body size in the empirical dataset ( Figure 5). However, in the model output, the length of the thoracic tergites decrease posteriorly in the trunk at all molt stages. In the empirical dataset, the length of the thoracic tergites tends to increase posteriorly for the first four or five segments before decreasingly to the pygidium (there is variation in this, but it is not clear how much might be due to measurement error), a pattern that has been seen in other trilobites as well.

Parameter variation and resulting relative growth
The relative length of different sclerites, as well as maximum body size, may be explored by varying different parameters: Influence of varying the initial cephalic and pygidial length. It is likely that there was variation in the size of the cephalon and transitory pygidium at M0. In order to capture the influence of this variation on total body length, the initial cephalic and pygidial lengths were sampled from normal distributions with means of 0.67 and 0.55, respectively, and standard deviations of 0.05. When the lengths of the cephalon and transitory pygidium at M0 are allowed to vary, the variance in body size at any given molt increases through growth, whereas the variance in body size in the empirical dataset is smaller for each molt, most notably at later molt stages( Figure 6). This puts a fine point on the fact that there is no canalization mechanism built into the current model (see Hughes et al. 2017 for discussion of growth canalization). At the latest molt stages, the median body size in the empirical dataset is consistently lower than the median from the model output. Although well within the range possible if initial cephalic and pygidial lengths varied, this suggests that growth rates may have slowed late in life in a way not captured by the current model.
Influence of varying the number of terminal tergites in the thorax. If no other parameters are changed, one might expect that reducing the number of tergites will result in smaller trilobites. This is true, but the decrease is small. For example, decreasing the number of terminal thoracic tergites by 50% (Ter =10) results in a trilobite that is only 85% smaller (Figure 7). Because the cephalon is growing at the same rate, it is the relative proportions of the trunk that vary to accommodate this. In particular, the relative length of the pygidium is much greater in a trilobite with fewer thoracic tergites because the pygidium is under positive growth (during which it no longer releases tergites into the thorax) for a greater proportion of the lifespan. In addition, any particular tergite numbered from the front (e.g., the second tergite) will undergo a relatively higher growth rate throughout the lifespan because it shares a larger portion of the growth gradient. However, smaller (and re-proportioned) trilobites would result if there was a limit on the total number of molts a trilobite underwent during holaspis (Molt < 31) rather than a limit on the total number of molts during the entire lifespan.
Influence of increasing rates systematically across body regions. It should be possible to generate a smaller or larger trilobite without altering body proportions if rates are adjusted systematically across all body regions. To determine the sensitivity of body size to changes in rates, the cephalic growth rate (g.c) and the minimum and maximum growth rates of the trunk region (lowM, highM, lowH, highH) were systematically adjusted and the model re-run. A four-percent increase in the rates across these five parameters results in a trilobite that is 3.5 times larger than before but with the same relative body proportions (Table 3). Influence of varying the growth gradient during meraspis or holaspis. In A. koninckii, the growth gradient becomes shallower at the transition from the meraspid period to the holaspid period, due to an increase in the minimum and decrease in the maximum rates ( Figure 3B).
The maximum rates along the growth gradient (highM, highH) primarily affect growth of the pygidium, and this transition also marks the point in growth where the pygidium actively increases in size because tergites are no longer being released into the thorax. Thus, one way to make an isopygous trilobite (one where the pygidium is of equal length to the cephalon) is to maintain the high growth rate at the posterior part of the trunk into holaspis, for example by retaining the meraspid growth gradient through holaspis ( Figure 7). One interesting result is also that the length of the thoracic tergites decreases and then increases posteriorly in the latest molt stages. The pygidium will eventually outgrow the cephalon if λ is increased during the holaspid period, which would decrease the concavity of the exponential decay curve and result in higher growth rates across a larger area of the posterior part of the trunk.
The holapsis growth curve in A. koninckii is shallow enough along most of the trunk that the relative length of the thorax remains almost constant during holaspis (Figure 4, 7). As the range of the growth gradient is decreased, the exponential curve becomes more and more similar to a horizontal line. This ultimately results in true isomorphic growth along the trunk (i.e., constant rates rather than a growth gradient) (Figure 9). Because of the shape of the exponential decay curve, a constant rate near the lower end of the growth gradient (1.084 for A. koninckii) yields a total body length closest to that of actual specimens, but the relative length of the pygidium to the cephalon is smaller. Further, because the minimum growth rate of the trunk (lowH) is almost exactly the same as the cephalic growth rate, the relative proportions of the cephalon, thorax, and pygidium will remain constant during holaspis when the holaspid rates are constant and low ( Figure 9G-I). Influence of decoupling between timing of shift in growth gradients and termination of tergite release.
Because the growth gradient is shallower and has a smaller maximum rate during holaspis than during meraspis (at least in A. koninckii), altering the timing of transition between growth gradients has differential effects depending on whether the transition between growth gradients occurs before or after termination of tergite release ( Figure 10, Table 4). If the transition happens earlier in ontogeny relative to the termination of tergite release, the total exoskeletal length is slightly smaller because rates decrease sooner but the pygidium is much smaller because tergite release continues even though the pygidium grows at a slower rate. In contrast, if the transition happens later in ontogeny relative to the termination of tergite release, the total exoskeletal length is slightly larger but the pygidium is much larger because high growth rates continue even after tergites are no longer being released. The change in relative proportions would not be as great if new segment generation occurred in the thorax as opposed to the pygidium. How many specimens are needed to estimate growth parameter?
The Aulacopleura koninckii of dataset of Hughes et al. (2017) comprises 351 well-preserved articulated specimens, 40% of which span the ontogeny of the species from meraspid stage 4 to 17 (sample sizes for each stage shown in Supplementary Figure 2). Such a collection of specimens is exceedingly rare, for both taphonomic and practical reasons. Thus, it may be useful to know how many specimens are needed to accurately estimate growth parameters for this and other species, and what level of precision might be expected under different sampling scenarios. Subsampling routines of the A. koninckii dataset produced the following results.

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Deleted: 1 One specimen per stage. One specimen per stage (M9-M17) was randomly selected from the dataset and the following parameters were estimated: 1) the cephalic length at M0 (ce); 2) the transitory pygidium length at M0 (p); 3) the rate of growth of the cephalon (g.c); 4) the proportional size of the newly released tergite at each stage (t.p); and 5) the minimum (lowM), maximum (highM), and exponential decay constant (lambdaM) describing the growth gradient. This was repeated 1000 times. In each iteration, each stage was represented by one specimen.
All of the input parameters (ce, Figure 11A; p, Figure 11B; g.c, Figure 11C, t.p, Figure 11D) are well estimated, while the minimum and maximum growth rates in the trunk (lowM, Figure 11E, highM, Figure   11F) tend to be overestimated, and the exponential decay constant tends to be underestimated (lambdaM, Figure 11G). The result of this imprecision on input parameters is expected to be small, given that impact on total body size of varying the initial cephalic and pygidial length (ce and p) using a higher standard deviation ( Figure 6). However, the ranges of values for the growth gradient parameters is high: even excluding the extreme outliers, the range of values is greater than the difference between the meraspid and holaspid growth gradients estimated for A. koninckii (Table 1). Given the results of varying these parameters above (Figures 6-10), poor estimates for growth gradient parameters are expected have a greater impact on model output than poor estimates for input parameters.
More than one specimen per stage. n specimens were randomly selected from each of stages M9-M17, where n ranged from 2 to 8 (8 is the minimum number of specimens representing these stages, see Supplementary Figure 2), and the four input and three meraspid growth parameters were estimated. This was repeated 1000 times. In each iteration, each stage was represented by the same number of specimens.
For graphical comparison, univariate density estimates were constructed from the frequency distributions using the sm.density function in the R package sm ver. 2.2-5.6 (Bowman and Azzalini 2018).
As expected, with an increase in the number of specimens per stage, the range in parameter estimates decreased ( Figure 12). The mean estimates shifted towards values estimated from the entire empirical dataset, most notably for growth gradient parameters which were more severely over-or underestimated at low sample sizes ( Figure 12E-G). Here, where specimens were being drawn from the latter half of the meraspid period (M9-M17), at least six specimens per meraspid molt stage were needed to accurately estimate parameters describing the growth gradient. For both input and growth gradient parameters, sampling more than six specimens per meraspid growth stage yielded increasingly precise estimates, but with diminishing returns.
A few specimens from few to many stages. n specimens from k stages were randomly selected from the dataset and the four input and three meraspid growth parameters were estimated. For the input parameters, this was repeated 1000 times for each of k ranging from 3 to 14 (representing just three to all stages M4-M17) and for n number of specimens ranging from 3 to 5. Because determining the growth gradient requires a larger sample size (= larger number of molt stages represented), growth parameters were estimated for each of k ranging from 10 to 14 (representing ten to all stages M4-M17). In each iteration, each selected stage was represented by the same number of specimens.
As expected, with an increase in the number of stages represented, the range in parameter estimates decreased (Figure 13, 14). However, there was no shift in the mean estimate of growth gradient parameters as the number of stages represented was increased ( Figure 14). This implies that once many meraspid stages are represented in a sample (> 10), fewer specimens per stage are required to get accurate estimates. The precision was more quickly improved by increasing the number of stages represented than by increasing the number of specimens (within a small range) in each stage ( Figure 14).

Implications and applications of the model
With the approach proposed in this paper, a minimum of nine parameters are required to model basic trilobite growth and segmentation, and three additional parameters are required to allow a transition to a new growth gradient for the trunk region during ontogeny. It is possible to add additional parameters to describe additional variation in the segmentation schedule or variation in growth rates throughout ontogeny. As an example herein, the parameter R allows for variation in the timing of the transition to a new growth gradient. Other modifications could include varying the timing of tergite release in the thorax. For example, some trilobites alternated segment generation in the pygidium and segment release into the trunk (Hughes et al. 2006;Hopkins 2017).Finally, many species may have undergone multiple molt cycles while still in the protaspid period and this could lead to variation in body sizes and the proportion of cephalic to pygidial lengths at the start of the meraspid period. Although this model does not explicitly include the earliest sclerotized developmental stages, the impact of differential growth or growth variation in the protaspid period may be approximated by varying the ce and p input parameters, as described above ( Figure 6).
In addition to increasing the complexity of the model to better represent specific or unusual developmental patterns in trilobites, it should also be possible to apply this approach to modeling growth in other arthropods, particularly other hemianamorphic arthropods for which segmentation patterns are known (e.g., conchostracans, Oleson 1999, copepods, Ferrari andDahms 2007). One of the major obstacles in applying this to other arthropods may be the initial estimation of parameters from empirical data, which requires samples comprising multiple specimens from multiple molt stages (although good  Figure 1B). Because most pygidial growth occurs during holaspis, depressing the growth gradient during this period yields body proportions consistent with micropygous trilobites, where the pygidium is much smaller in length and width than the cephalon ( Figure 9G-I), as seen in many olenellid trilobites (Supplementary Figure 1C).
However, similar body proportions may be produced when the transition to the holaspid growth gradient is initiated before termination of thoracic tergite release ( Figure 10A-C) or when cephalic growth rates are increased relative to the growth gradient across the trunk (not shown). Thus it is possible that specific combinations of body size and proportions may be attainable by more than one combination of model parameters. Most trilobites have body proportions that vary between the micropygous and isopygous conditions (Supplementary Figure 1D-F).
Given a set number of molts, small changes in growth rates can have large effects on total body size or relative body proportions. The extent to which such changes (and of this magnitude) are ecologically or evolutionarily possible is an open question. A 50% increase in body size is seen in Poseidonmicus ostracods over the last 40 million years in association with a decrease in ocean bottom-water temperatures, and some species exhibit almost this entire range of body size across historic populations (Hunt and Roy 2006). Because ostracods have determinant growth and a fixed number of molt stages, it seems likely that this body size variation was achieved by changes to per-molt growth rates. However, in arthropods with indeterminate growth (like trilobites), variation in adult body sizes in response to environmental variation (e.g., temperature, food availability, presence of predators) is often achieved by some combination of changes in magnitude of per-molt increase in body size and changes in molt frequency and/or number of molts per life history phase (Maszczyk and Brzezinski 2018). Better understanding of growth response to environmental variation in trilobites requires further empirical studies (e.g., comparisons of growth rates in environmentally-separated populations of the same species).
In the meantime, the model proposed herein may provide a framework for assessing the range of realized body sizes and body proportions in trilobites, and for revealing possible ways in which body size and body proportions may have evolved.